Evaluate triple integral, involves e -(x 2)

In summary, the conversation discusses how to evaluate the triple integral of e**-(x**2 + 2y**2 + 3z**2) with limits from -infinity to infinity. The individual trying to solve the problem is unsure of which coordinate system to use and considers using cylindrical or spherical coordinates. Another individual suggests using rectangular coordinates and mentions that the integrand is in the form of f(x)g(y)h(z). The conversation then shifts to discussing the integral \displaystyle \int_{-\infty}^{+\infty}e^{-u^2}du\ and whether or not it is necessary to know for solving the problem in cartesian coordinates. The link to the Gaussian integral is provided as a resource.
  • #1
MeMoses
129
0
Evaluate triple integral, involves e**-(x**2)

Homework Statement


Evaluate the triple integral of e**-(x**2 + 2y**2 + 3z**2), all of the limits are from -infinity to infinity.


Homework Equations





The Attempt at a Solution


I'm not really sure how to do this problem. I know I have to change the coordinate system, but what to? I tried cylindricals, but it didn't seem right, unless I made a mistake there. Or do i have have to use spherical coordinates, which I am reading up on right now. Any help would be great.
 
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  • #2


MeMoses said:

Homework Statement


Evaluate the triple integral of e**-(x**2 + 2y**2 + 3z**2), all of the limits are from -infinity to infinity.

The Attempt at a Solution


I'm not really sure how to do this problem. I know I have to change the coordinate system, but what to? I tried cylindricals, but it didn't seem right, unless I made a mistake there. Or do i have have to use spherical coordinates, which I am reading up on right now. Any help would be great.
I would do it in rectangular coordinates. The integrand is of the form, f(x)g(y)h(z).

Do you know the result for [itex]\displaystyle \int_{-\infty}^{+\infty}e^{-u^2}du\ ?[/itex]
 
  • #3


I do not, and I imagine I will need some proof for my values so I don't think it's possible in cartesian coordinates unless I know the integral you stated. Is there an easy way to show proof for that integral?
 
  • #4


MeMoses said:
I do not, and I imagine I will need some proof for my values so I don't think it's possible in cartesian coordinates unless I know the integral you stated. Is there an easy way to show proof for that integral?

http://en.wikipedia.org/wiki/Gaussian_integral
 

FAQ: Evaluate triple integral, involves e -(x 2)

What is a triple integral?

A triple integral is an integral that involves integrating a function over a three-dimensional region. It is represented by three nested integrals and is used to calculate the volume of a solid bounded by three surfaces, or the mass, center of mass, or moment of inertia of a three-dimensional object.

What does "e -(x 2)" represent in the triple integral?

The "e -(x 2)" in the triple integral represents the function that is being integrated. In this case, it is a function of x that involves the natural number e raised to the power of negative x squared.

How do you evaluate a triple integral?

To evaluate a triple integral, you first need to set up the limits of integration for each variable (x, y, and z) based on the given region. Then, you can use the properties of integration and the appropriate integration techniques (such as substitution or integration by parts) to solve each of the nested integrals. Finally, you can combine the results to get the value of the triple integral.

What is the significance of "e" in the triple integral?

The natural number e, approximately equal to 2.718, is a mathematical constant that appears in many areas of mathematics, including calculus. In the context of a triple integral, "e" may represent a rate of growth or decay, or it may be a constant factor in the function being integrated.

In what real-life situations would a triple integral be used?

Triple integrals have various applications in physics, engineering, and other fields. For example, they can be used to calculate the volume or mass of a three-dimensional object, the flux of a vector field, or the work done by a force in a three-dimensional system. They are also used in the process of finding the center of mass, moment of inertia, or other physical properties of a three-dimensional object.

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